I want to calculate the Ricci scalar using the Cartan equations in the context of reducing a spatial dimension of a 4-dimensional theory describing an empty spacetime with a metric of the form \begin{equation} d \hat{s} = e^{2 \alpha \varphi} d s_{3}^{2} + e^{2 \beta \varphi} d \phi^{2} \end{...

I consider free energy functional at transition point \begin{equation} \mathcal{F} = \int d^4 r \big[\frac{b}2 (\nabla \phi)^2 + \frac{\lambda}{24} (\phi^2)^2 \big], \end{equation} where $\phi$ is 4 component order parameter \begin{equation} \phi^2 = \sum_{i=1}^4 \phi^2_i \qquad (\nabla \phi)^2 =...

This is a general question regarding the motion of fragments of a single particle at rest, with no external force acting on it. Say I have a (point like) body of mass $m$, situated at the origin of my coordinate system. Due to some internal forces, if the body happens to split into two symmetrica...

Given probe that falls into BH (simple non-moving mass M) staring from distance r=D, velocity v=0 and clock t=0, what will the clock show when entering the EH (or in general as function of the distance from the BH)? I found (e.g. in wiki) formulas for objects not moving and with circular orbits b...